A002515 -id:A002515 - cp's OEIS frontend

A122095 Primes p for which 8*p+1 divides 2^p-1.

11, 29, 179, 239, 431, 761, 857, 941, 1367, 1667, 1871, 1877, 2411, 2837, 3041, 3119, 3329, 3347, 3767, 4289, 5021, 5087, 5231, 5261, 5717, 5861, 6449, 6917, 6959, 7079, 7211, 7919, 8429, 8741, 8867, 9341, 9461, 9851, 10211, 10979, 12107, 12437, 12479

Offset: 1

J. Lowell, Oct 17 2006

nonn

The first 962 terms, all those with n<500000, are also in A023228. - R. J. Mathar, Oct 20 2006

All terms are in A023228, i.e., such that 8p+1 is prime, since a divisor of 8p+1 would also divide M(p)=A000225(p) and thus be of the form 2kp+1, but it is easily checked that 8p+1 cannot be a multiple of 2p+1 (nor of 4p+1 or 6p+1, of course). - M. F. Hasler, Mar 21 2011

			29 is in this sequence because 2^29-1 is divisible by 8 * 29 + 1 = 233.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000225, A002515, A188130.

isA122095 := proc(n) RETURN( isprime(n) and ( (2^n-1) mod (8*n+1)) = 0 ) ; end: n := 1 : for a from 2 to 500000 do if isA122095(a) then print(n,a) ; n := n+1 ; fi ; od ; # R. J. Mathar, Oct 20 2006

Select[Prime[Range[1500]],Divisible[2^#-1,8#+1]&] (* Harvey P. Dale, Dec 18 2012 *)
Select[Prime[Range[1500]],PowerMod[2,#,8#+1]==1&] (* Harvey P. Dale, May 28 2015 *)

forprime( p=1,1e4, Mod(2,p*8+1)^p-1 || print1(p, ", "))

More terms from R. J. Mathar, Oct 20 2006

A155151 Triangle T(n, k) = 4nk + 2n + 2k + 2, read by rows.

Original entry on oeis.org

10, 16, 26, 22, 36, 50, 28, 46, 64, 82, 34, 56, 78, 100, 122, 40, 66, 92, 118, 144, 170, 46, 76, 106, 136, 166, 196, 226, 52, 86, 120, 154, 188, 222, 256, 290, 58, 96, 134, 172, 210, 248, 286, 324, 362, 64, 106, 148, 190, 232, 274, 316, 358, 400, 442, 70, 116, 162

Offset: 1

Vincenzo Librandi, Jan 21 2009

First column: A016957, second column: A017341, third column: 2*A017029, fourth column: A082286. - Vincenzo Librandi, Nov 21 2012
Conjecture: Let p = prime number. If 2^p belongs to the sequence, then 2^p-1 is not a Mersenne prime. - Vincenzo Librandi, Dec 12 2012
Conjecture is true because if T(n, k) = 2^p with p prime, then 2^p-1 = 4*n*k + 2*n + 2*k + 1 = (2*n+1)*(2*k+1) hence 2^p-1 is not prime. - Michel Marcus, May 31 2015
It appears that T(m,p) = 2^p for Lucasian primes (A002515) greater than 3. For instance: T(44, 11) = 2^11, T(89240, 23) = 2^23. - Michel Marcus, May 28 2015
For n > 1, ascending numbers along the diagonal are also terms of the even principal diagonal of a 2n X 2n spiral (A137928). - Avi Friedlich, May 21 2015

			Triangle begins
  10;
  16,  26;
  22,  36,  50;
  28,  46,  64,  82;
  34,  56,  78, 100, 122;
  40,  66,  92, 118, 144, 170;
  46,  76, 106, 136, 166, 196, 226;
  52,  86, 120, 154, 188, 222, 256, 290;
  58,  96, 134, 172, 210, 248, 286, 324, 362;
  64, 106, 148, 190, 232, 274, 316, 358, 400, 442;

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A000043, A000668, A016957, A017029, A017341, A054723, A082286, A144650.

[4*n*k + 2*n + 2*k + 2: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012

seq(seq( 2*(2*n*k+n+k+1), k=1..n), n=1..15) # G. C. Greubel, Mar 21 2021

T[n_,k_]:=4*n*k + 2*n + 2*k + 2; Table[T[n, k], {n, 11}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

flatten([[2*(2*n*k+n+k+1) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 21 2021

T(n, k) = 2*A144650(n, k).

Sum_{k=1..n} T(n,k) = n*(2*n^2 + 5*n + 3) = n*A014105(n+2) =

Edited by Robert Hochberg, Jun 21 2010

A188133 Primes p such that 10p+1 divides 2^p-1.

Original entry on oeis.org

43, 487, 547, 571, 883, 1459, 1663, 1723, 2539, 3319, 3511, 4903, 5107, 5431, 6199, 6367, 8011, 8599, 9007, 9391, 9511, 10111, 11119, 11959, 12379, 12703, 13291, 13339, 13999, 14083, 14551, 14767, 15187, 15319, 15859, 15991, 16183, 16603, 16747, 17659, 18427, 19699

Offset: 1

M. F. Hasler, Mar 21 2011

nonn

It is known that divisors of M(p)=2^p-1 are of the form 2kp+1. For k=1, these are the Lucasian primes A002515, for k=2 there are no such divisors, for k=3 these divisors are listed in A188130 and for k=4 in A122095.
The equivalent sequence of prime indices is 14, 93, 101, 105, 153, 232, 261, 269, ....
If k == 2 (mod 4), there are no such divisors in general and here there are no smaller k's than k = 5. - Karl-Heinz Hofmann, Jan 27 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002515 (k = 1), A188130 (k = 3), A122095 (k = 4), A350702 (k = 7).

Cf. A001348, A000040.

Select[Range[2*10^4], PrimeQ[#] && PowerMod[2, #, 10# + 1] == 1 &] (* Amiram Eldar, Nov 13 2019 *)
Select[Prime[Range[2500]],PowerMod[2,#,10#+1]==1&] (* Harvey P. Dale, Dec 08 2024 *)

forprime(p=1,1e5, Mod(2,p*10+1)^p-1 || print1(p", "))

from sympy import sieve
print([p for p in sieve[1:10000] if pow(2,p,10*p+1) == 1])
# Karl-Heinz Hofmann, Jan 27 2022

{p = A000040(i): 10*p+1 | A001348(i)}. - R. J. Mathar, Mar 21 2011

A374913 Numbers k such that k^(k + 1) == k + 1 (mod 2*k + 1).

Original entry on oeis.org

2, 3, 6, 11, 14, 15, 18, 23, 26, 30, 35, 39, 50, 51, 54, 63, 74, 75, 78, 83, 86, 90, 95, 98, 99, 111, 114, 119, 131, 134, 135, 138, 146, 155, 158, 174, 179, 183, 186, 191, 194, 198, 210, 215, 219, 230, 231, 239, 243, 251, 254, 270, 278, 299, 303, 306, 315, 323, 326, 330, 338, 350

Offset: 1

Juri-Stepan Gerasimov, Jul 23 2024

nonn

Michel Marcus, Table of n, a(n) for n = 1..10000

Supersequence of A002515 and A374914.

Cf. A374912.

[n: n in [0..350] | n^(n+1) mod (2*n+1) eq n+1];

Select[Range[350],Mod[#^(#+1),2#+1]==#+1 &] (* Stefano Spezia, Jul 23 2024 *)

isok(k) = Mod(k, 2*k+1)^(k+1) == k+1; \\ Michel Marcus, Feb 05 2025

Conjecture (Superseeker): a(n) = A263458(n)/2. - R. J. Mathar, Aug 02 2024

The conjectured formula is false. There exist numbers k such that 2*k + 1 is composite and k^(k + 1) == k + 1 (mod 2*k + 1). For example, when k = 1023: 1023^1024 == 1024 (mod 2047) and 2047 = 23*89 is composite. - Jedrzej Miarecki, Jan 16 2025

A186522 Smallest prime factor of 2^n - 1 having the form k*n + 1.

Original entry on oeis.org

3, 7, 5, 31, 7, 127, 17, 73, 11, 23, 13, 8191, 43, 31, 17, 131071, 19, 524287, 41, 127, 23, 47, 241, 601, 2731, 262657, 29, 233, 31, 2147483647, 257, 599479, 43691, 71, 37, 223, 174763, 79, 41, 13367, 43, 431, 89, 631, 47, 2351, 97, 4432676798593, 251, 103, 53, 6361, 87211, 881, 113, 32377, 59, 179951, 61

Offset: 2

T. D. Noe, Feb 23 2011

nonn

The values of k are in A186283.
From Zhi-Wei Sun, Dec 27 2016: (Start)
For any odd prime p, by Fermat's little theorem p = (p-1) + 1 divides 2^(p-1) - 1, and it is well-known that any prime divisor q of 2^p - 1 must be congruent to 1 modulo p.
Conjecture: a(n) exists for any integer n > 1 (verified for n = 2..300). (End)
Proof of the above conjecture: By Bang's theorem, for each n > 1 except 6 there exists an odd prime p such that the multiplicative order of 2 modulo p is n, and therefore n must divide p-1. Note that a(n) <= p. - Robert Israel and Thomas Ordowski, Sep 08 2017
For prime p, a(p) = 2p + 1 if and only if p is a Lucasian prime (A002515). - Thomas Ordowski, Sep 08 2017

			For n = 4, the prime factors of 2^n - 1 are 3 and 5, but only 5 has the form k * n + 1. Hence a(4) = 5.
a(254) = 56713727820156410577229101238628035243 since this prime number is equal to (2^127+1)/3 and congruent to 1 modulo 127, and 2^127 - 1 is a Mersenne prime.
a(257) = 535006138814359 since this is a prime congruent to 1 modulo 257 and 2^257 - 1 = 535006138814359*p*q with p = 1155685395246619182673033 and q = 374550598501810936581776630096313181393 both prime. - _Zhi-Wei Sun_, Dec 27 2016

Table of n, a(n) for n = 2..1236
GIMPS (Prime95), Mersenne exponent cofactor probable prime PRP.

Cf. A000040, A000225, A060443 (all prime factors of 2^n-1).

Table[p = First/@FactorInteger[2^n - 1]; Select[p, Mod[#1, n] == 1 &, 1][[1]], {n, 2, 60}]

a(n)=my(s=if(n%2,2*n,n));forstep(p=s+1,2^n-1,s, if(Mod(2,p)^n==1&&isprime(p), return(p))) \\ Charles R Greathouse IV, Sep 07 2017

a(n)=my(f=factor(2^n-1)[,1]); for(i=1,#f, if(f[i]%n==1, return(f[i]))) \\ Charles R Greathouse IV, Sep 07 2017

a(p - 1) = p for odd prime p. - Thomas Ordowski, Sep 04 2017
A002326((a(n)-1)/2) divides n for all n > 1. - Thomas Ordowski, Sep 07 2017
a(n) = A186283(n) * n + 1. - Max Alekseyev, Apr 27 2022

Terms to a(300) in b-file from Zhi-Wei Sun, Dec 27 2016
a(301)-a(1200) in b-file from Charles R Greathouse IV, Sep 07 2017
a(1201)-a(1236) in b-file from Max Alekseyev, Apr 27 2022

A230809 Primes p of the form 60n + 59 such that 2p + 1 is also prime.

Original entry on oeis.org

179, 239, 359, 419, 659, 719, 1019, 1439, 1499, 1559, 2039, 2339, 2399, 2459, 2699, 2819, 2939, 3299, 3359, 3539, 3779, 4019, 4919, 5039, 5279, 5399, 5639, 6899, 7079, 9059, 9419, 9479, 9539, 10799, 11519, 11579, 11699, 11939, 12119, 12899, 12959, 13619

Offset: 1

Arkadiusz Wesolowski, Oct 30 2013

nonn

Primes p such that 2*p + 1 divides Lucas(p) and Mersenne(p).

			179 is in the sequence since it is prime and 359 is a factor of both Lucas(179) and Mersenne(179) = 2^179 - 1.

Eric Weisstein's World of Mathematics, Lucas Number
Eric Weisstein's World of Mathematics, Mersenne Number

Subsequence of A142799, of A215850, and of A239548. Cf. A000032, A001348, A002515.

[p : p in [59..13619 by 60] | IsPrime(p) and IsPrime(2*p+1)];

forstep(p=59, 13619, 60, if(isprime(p)&&isprime(2*p+1), print1(p, ", ")));

A005384 INTERSECT A142799.

A002515 INTERSECT A215850.

A291691 Primes p such that gpf(lpf(2^p - 1) - 1) = p.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 23, 29, 37, 43, 47, 53, 73, 79, 83, 97, 113, 131, 151, 173, 179, 181, 191, 197, 211, 223, 233, 239, 251, 263, 277, 281, 283, 307, 317, 337, 353, 359, 367, 383, 397, 419, 431, 439, 443, 457, 461, 463, 467, 487, 491, 499

Offset: 1

Thomas Ordowski, Aug 30 2017

nonn

This sequence has not been proved to be infinite.
The terms p such that 2^p - 1 is a Mersenne prime are 2, 3, 5, 7, and 13.
If p is prime, then gpf(lpf(2^p - 1) - 1) >= p.
Primes q such that gpf(lpf(2^q - 1) - 1) > q are A292237.

			We have gpf(lpf(2^11 - 1) - 1) = gpf(23 - 1) = 11, so 11 is a term.

Charles R Greathouse IV, Table of n, a(n) for n = 1..119

Cf. A000040, A000043, A006530, A016047, A020639, A236128.

Contains A002515.

lpf[n_] := FactorInteger[n][[1, 1]]; gpf[n_] := FactorInteger[n][[-1, 1]]; Select[ Prime@ Range@ 45, gpf[lpf[2^# - 1] - 1] == # &] (* Giovanni Resta, Aug 30 2017 *)

listp(nn) = forprime(p=2, nn, if (vecmax(factor(vecmin(factor(2^p-1)[,1])-1)[,1]) == p, print1(p, ", "));); \\ Michel Marcus, Aug 30 2017

a(17)-a(26) from Michel Marcus, Aug 30 2017
a(27)-a(34) from Giovanni Resta, Aug 30 2017
a(35)-a(52) from Charles R Greathouse IV, Aug 30 2017

A350702 Primes p such that 14*p + 1 divides 2^p - 1.

Original entry on oeis.org

929, 1433, 2393, 2609, 2657, 4373, 4793, 6029, 7529, 10133, 10433, 10949, 10973, 13049, 13109, 16829, 18869, 20873, 22349, 23417, 24137, 26717, 27737, 27893, 28433, 28517, 30977, 33809, 33857, 37217, 38189, 38237, 39209, 39749, 41453, 41813, 42569, 43313, 43613

Offset: 1

Karl-Heinz Hofmann, Jan 27 2022

nonn

Known divisors of Mersenne(p) (2^p-1 or Mp for short) are of the form 2*k*p+1. See crossrefs for other k's. If k == 2 (mod 4), there are no such divisors in general. Here k is 14/2 = 7.

			See LINKS for example of a(13).

Karl-Heinz Hofmann, Mersenne(p) table with k = 7 and, if they exist, additional k < 7 plus their corresponding factors.
mersenne.ca, a(13) = M10973 Mersenne number exponent details.
mersenne.ca, a(1..995) at GIMPS with k = 7 and Exponent 3..2000000.

Cf. A002515 (k = 1), A188130 (k = 3), A122095 (k = 4), A188133 (k = 5).

Cf. A000040, A001348.

Select[Range[45000], PrimeQ[#] && PowerMod[2, #, 14*# + 1] == 1 &] (* Amiram Eldar, Jan 27 2022 *)

forprime(p=1, 1e6, if (Mod(2, p*14+1)^p==1, print1(p,", ")))

from sympy import sieve
print([p for p in sieve[1:1000000] if pow(2, p, 14*p+1) == 1])

{p = A000040(i): 14*p+1 | A001348(i)}.

A101320 Numbers k such that 4k-1, 8k-1, 16k-1, 32k-1, 64k-1 and 128k-1 are all primes.

Original entry on oeis.org

15855, 31785, 267300, 280665, 399675, 561330, 946050, 990510, 1022220, 1082115, 1164735, 1283250, 1303875, 1309545, 1514880, 1669065, 1924410, 2850225, 3078675, 3092760, 3492270, 3536385, 3611205, 3920670, 4148970, 4454775

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 23 2004

			4*15855-1, 8*15855-1, 16*15855-1, 32*15855-1, 64*15855-1 and 128*15855-1 are primes, so 15855 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Iain Fox)

Cf. A002515.

Subsequence of A005099, A005122, A101790, A101794 and A101994.

Select[Range[10^6], And @@ PrimeQ[2^Range[2, 7]*# - 1] &] (* Amiram Eldar, May 23 2024 *)

lista(nn) = for(n=1, nn, if(ispseudoprime(4*n-1) && ispseudoprime(8*n-1) && ispseudoprime(16*n-1) && ispseudoprime(32*n-1) && ispseudoprime(64*n-1) && ispseudoprime(128*n-1), print1(n, ", "))) \\ Iain Fox, Nov 23 2017

A101789 Safe primes of the form 8k-1: primes of the form 8k-1 such that 4*k-1 is also a prime.

Original entry on oeis.org

7, 23, 47, 167, 263, 359, 383, 479, 503, 719, 839, 863, 887, 983, 1319, 1367, 1439, 1487, 1823, 2039, 2063, 2207, 2447, 2879, 2903, 2999, 3023, 3119, 3167, 3623, 3863, 4007, 4079, 4127, 4679, 4703, 4799, 4919, 5087, 5399, 5639, 5807, 5879, 5927, 6047

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

			4*1-1 = 3 and 8*1-1 = 7 are primes, so the first term is 7.

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Intersection of A005385 and A007522.

Cf. A002515.

Select[Prime[Range[800]],Mod[#,8]==7&&PrimeQ[(#-1)/2]&] (* Harvey P. Dale, Jan 31 2012 *)

is(k) = (k % 8 == 7) && isprime(k) && isprime(k\2); \\ Amiram Eldar, May 23 2024

cp's OEIS Frontend

A122095 Primes p for which 8*p+1 divides 2^p-1.

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A155151 Triangle T(n, k) = 4*n*k + 2*n + 2*k + 2, read by rows.

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A188133 Primes p such that 10p+1 divides 2^p-1.

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A374913 Numbers k such that k^(k + 1) == k + 1 (mod 2*k + 1).

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Magma

Mathematica

PARI

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A186522 Smallest prime factor of 2^n - 1 having the form k*n + 1.

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Mathematica

PARI

PARI

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A230809 Primes p of the form 60*n + 59 such that 2*p + 1 is also prime.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

PARI

Formula

A291691 Primes p such that gpf(lpf(2^p - 1) - 1) = p.

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A155151 Triangle T(n, k) = 4nk + 2n + 2k + 2, read by rows.

A230809 Primes p of the form 60n + 59 such that 2p + 1 is also prime.

A101320 Numbers k such that 4k-1, 8k-1, 16k-1, 32k-1, 64k-1 and 128k-1 are all primes.

A101789 Safe primes of the form 8k-1: primes of the form 8k-1 such that 4*k-1 is also a prime.